Tham khảo tài liệu 'an introduction to financial option valuation_3', tài chính - ngân hàng, tài chính doanh nghiệp phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Computing asset paths 65 Fig. . Two discrete asset paths of the form . Lower picture has higher volatility. Fig. . Upper picture 20 discrete asset paths. Lower picture sample mean of 104 discrete asset paths. 66 Asset price model Part II Fig. . Upper picture 50 discrete asset paths over 0 T with 5b 1 p Ơ T 1 and St 10 2. Lower picture histogram for S T from 104 such paths with lognormal density function superimposed. it is visually indistinguishable from the exact mean s0ept that we derived in . We next give a test that confirms the lognormal behaviour of the asset model. Computational example Here we set 50 1 p and Ơ and computed discrete paths over 0 T with T 1. We used a uniform time spacing of ti 1 ti St 10 2. The upper picture in Figure shows 50 such paths. In the lower picture we give a kernel density estimate for the asset price at expiry. This was computed in the manner discussed in Section using a histogram with 45 bins of width . The corresponding lognormal density function which is superimposed as a dashed line gives a good match. Timescale invariance The next computational example reveals a key property of the asset price model. The jaggedness looks the same over a range of different timescales. In other words zooming in or out of the picture we see the same qualitative behaviour. We saw the same effect when we moved from daily to weekly data in Figures and . Timescale invariance 67 Fig. . The same asset path sampled at different scales. Upper picture 100 samples over 0 1 . Middle picture 100 samples over 0 . Lower picture 100 samples over 0 . Computational example To generate Figure we computed a single asset path for s0 1 J and Ơ at equally spaced time points in 0 1 a distance 10-4 apart. Using this data we plot three pictures. Each picture shows the path at 100 equally spaced time points. The upper plot shows the path at 100 equally spaced points in
